ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpdom2g Unicode version

Theorem xpdom2g 6732
Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom2g  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )

Proof of Theorem xpdom2g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xpeq1 4559 . . . . 5  |-  ( x  =  C  ->  (
x  X.  A )  =  ( C  X.  A ) )
2 xpeq1 4559 . . . . 5  |-  ( x  =  C  ->  (
x  X.  B )  =  ( C  X.  B ) )
31, 2breq12d 3948 . . . 4  |-  ( x  =  C  ->  (
( x  X.  A
)  ~<_  ( x  X.  B )  <->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
43imbi2d 229 . . 3  |-  ( x  =  C  ->  (
( A  ~<_  B  -> 
( x  X.  A
)  ~<_  ( x  X.  B ) )  <->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) ) )
5 vex 2692 . . . 4  |-  x  e. 
_V
65xpdom2 6731 . . 3  |-  ( A  ~<_  B  ->  ( x  X.  A )  ~<_  ( x  X.  B ) )
74, 6vtoclg 2749 . 2  |-  ( C  e.  V  ->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
87imp 123 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   class class class wbr 3935    X. cxp 4543    ~<_ cdom 6639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fv 5137  df-dom 6642
This theorem is referenced by:  xpdom1g  6733
  Copyright terms: Public domain W3C validator